435 lines
18 KiB
Text
435 lines
18 KiB
Text
/-
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Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Joseph Tooby-Smith
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-/
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import Mathlib.LinearAlgebra.UnitaryGroup
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import Mathlib.Data.Complex.Exponential
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import Mathlib.Tactic.Polyrith
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/-!
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# The CKM Matrix
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The definition of the type of CKM matrices as unitary $3×3$-matrices.
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An equivalence relation on CKM matrices is defined, where two matrices are equivalent if they are
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related by phase shifts.
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The notation `[V]ud` etc can be used for the elements of a CKM matrix, and
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`[V]ud|us` etc for the ratios of elements.
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-/
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open Matrix Complex
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noncomputable section
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/-- Given three real numbers `a b c` the complex matrix with `exp (I * a)` etc on the
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leading diagonal. -/
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@[simp]
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def phaseShiftMatrix (a b c : ℝ) : Matrix (Fin 3) (Fin 3) ℂ :=
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![![cexp (I * a), 0, 0], ![0, cexp (I * b), 0], ![0, 0, cexp (I * c)]]
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/-- The phase shift matrix for zero-phases is the identity. -/
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lemma phaseShiftMatrix_one : phaseShiftMatrix 0 0 0 = 1 := by
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ext i j
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fin_cases i <;> fin_cases j
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any_goals rfl
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· simp only [phaseShiftMatrix, ofReal_zero, mul_zero, exp_zero, Fin.zero_eta, Fin.isValue,
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cons_val', cons_val_zero, empty_val', cons_val_fin_one, vecCons_const, one_apply_eq]
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· simp only [phaseShiftMatrix, ofReal_zero, mul_zero, exp_zero, Fin.mk_one, Fin.isValue,
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cons_val', cons_val_one, head_cons, empty_val', cons_val_fin_one, one_apply_eq]
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· simp only [phaseShiftMatrix, ofReal_zero, mul_zero, exp_zero, Fin.reduceFinMk, cons_val',
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Fin.isValue, cons_val_two, Nat.succ_eq_add_one, Nat.reduceAdd, tail_cons, head_cons, empty_val',
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cons_val_fin_one, head_fin_const, one_apply_eq]
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/-- The conjugate transpose of the phase shift matrix is the phase-shift matrix
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with negated phases. -/
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lemma phaseShiftMatrix_star (a b c : ℝ) :
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(phaseShiftMatrix a b c)ᴴ = phaseShiftMatrix (- a) (- b) (- c) := by
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funext i j
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fin_cases i <;> fin_cases j <;>
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simp only [phaseShiftMatrix, Fin.zero_eta, Fin.isValue, conjTranspose_apply, cons_val',
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cons_val_zero, empty_val', cons_val_fin_one, vecCons_const, star_def, ← exp_conj,
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_root_.map_mul, conj_I, conj_ofReal, neg_mul, ofReal_neg, mul_neg, Fin.mk_one, cons_val_one,
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head_fin_const, star_zero, head_cons, cons_val_two, Nat.succ_eq_add_one, Nat.reduceAdd,
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tail_val', head_val', tail_cons, Fin.reduceFinMk, map_eq_zero]
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· rfl
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· rfl
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/-- The multiple of two phase shift matrices is equal to the phase shift matrix with
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added phases. -/
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lemma phaseShiftMatrix_mul (a b c d e f : ℝ) :
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phaseShiftMatrix a b c * phaseShiftMatrix d e f = phaseShiftMatrix (a + d) (b + e) (c + f) := by
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ext i j
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fin_cases i <;> fin_cases j <;>
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simp only [phaseShiftMatrix, Fin.zero_eta, Fin.isValue, mul_apply, cons_val', empty_val',
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cons_val_fin_one, cons_val_zero, vecCons_const, Fin.sum_univ_three, cons_val_one, head_cons,
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head_fin_const, mul_zero, add_zero, cons_val_two, Nat.succ_eq_add_one, Nat.reduceAdd, tail_cons,
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tail_val', head_val', zero_mul, ofReal_add, Fin.mk_one, Fin.reduceFinMk, zero_add]
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any_goals rw [mul_add, exp_add]
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change cexp (I * ↑c) * 0 = 0
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simp
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/-- Given three real numbers `a b c` the unitary matrix with `exp (I * a)` etc on the
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leading diagonal. -/
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@[simps!]
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def phaseShift (a b c : ℝ) : unitaryGroup (Fin 3) ℂ :=
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⟨phaseShiftMatrix a b c,
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by
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rw [mem_unitaryGroup_iff]
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change _ * (phaseShiftMatrix a b c)ᴴ = 1
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rw [phaseShiftMatrix_star, phaseShiftMatrix_mul, ← phaseShiftMatrix_one]
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simp only [phaseShiftMatrix, add_neg_cancel, ofReal_zero, mul_zero, exp_zero]⟩
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/-- The underlying matrix of the phase-shift element of the unitary group is the
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phase-shift matrix. -/
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lemma phaseShift_coe_matrix (a b c : ℝ) : ↑(phaseShift a b c) = phaseShiftMatrix a b c := rfl
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/-- The relation on unitary matrices (CKM matrices) satisfied if two unitary matrices
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are related by phase shifts of quarks. -/
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def PhaseShiftRelation (U V : unitaryGroup (Fin 3) ℂ) : Prop :=
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∃ a b c e f g, U = phaseShift a b c * V * phaseShift e f g
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/-- The relation `PhaseShiftRelation` is reflective. -/
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lemma phaseShiftRelation_refl (U : unitaryGroup (Fin 3) ℂ) : PhaseShiftRelation U U := by
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use 0, 0, 0, 0, 0, 0
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rw [Subtype.ext_iff_val]
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simp only [Submonoid.coe_mul, phaseShift_coe_matrix, ofReal_zero, mul_zero, exp_zero]
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rw [phaseShiftMatrix_one]
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simp only [one_mul, mul_one]
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/-- The relation `PhaseShiftRelation` is symmetric. -/
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lemma phaseShiftRelation_symm {U V : unitaryGroup (Fin 3) ℂ} :
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PhaseShiftRelation U V → PhaseShiftRelation V U := by
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intro h
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obtain ⟨a, b, c, e, f, g, h⟩ := h
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use (- a), (- b), (- c), (- e), (- f), (- g)
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rw [Subtype.ext_iff_val]
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rw [h]
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repeat rw [mul_assoc]
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simp only [Submonoid.coe_mul, phaseShift_coe_matrix, ofReal_neg, mul_neg]
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rw [phaseShiftMatrix_mul]
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repeat rw [← mul_assoc]
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simp only [phaseShiftMatrix_mul, neg_add_cancel, phaseShiftMatrix_one, one_mul, add_neg_cancel,
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mul_one]
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/-- The relation `PhaseShiftRelation` is transitive. -/
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lemma phaseShiftRelation_trans {U V W : unitaryGroup (Fin 3) ℂ} :
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PhaseShiftRelation U V → PhaseShiftRelation V W → PhaseShiftRelation U W := by
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intro hUV hVW
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obtain ⟨a, b, c, e, f, g, hUV⟩ := hUV
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obtain ⟨d, i, j, k, l, m, hVW⟩ := hVW
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use (a + d), (b + i), (c + j), (e + k), (f + l), (g + m)
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rw [Subtype.ext_iff_val, hUV, hVW]
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simp only [Submonoid.coe_mul, phaseShift_coe_matrix]
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rw [mul_assoc, mul_assoc, phaseShiftMatrix_mul, ← mul_assoc, ← mul_assoc, phaseShiftMatrix_mul,
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add_comm k e, add_comm l f, add_comm m g]
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/-- The relation `PhaseShiftRelation` is an equivalence relation. -/
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lemma phaseShiftRelation_equiv : Equivalence PhaseShiftRelation where
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refl := phaseShiftRelation_refl
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symm := phaseShiftRelation_symm
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trans := phaseShiftRelation_trans
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/-- The type of CKM matrices. -/
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def CKMMatrix : Type := unitaryGroup (Fin 3) ℂ
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/-- Two CKM matrices are equal if their underlying unitary matrices are equal. -/
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lemma CKMMatrix_ext {U V : CKMMatrix} (h : U.val = V.val) : U = V := by
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cases U
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cases V
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subst h
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rfl
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/-- The `ud`th element of the CKM matrix. -/
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scoped[CKMMatrix] notation (name := ud_element) "[" V "]ud" => V.1 0 0
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/-- The `us`th element of the CKM matrix. -/
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scoped[CKMMatrix] notation (name := us_element) "[" V "]us" => V.1 0 1
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/-- The `ub`th element of the CKM matrix. -/
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scoped[CKMMatrix] notation (name := ub_element) "[" V "]ub" => V.1 0 2
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/-- The `cd`th element of the CKM matrix. -/
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scoped[CKMMatrix] notation (name := cd_element) "[" V "]cd" => V.1 1 0
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/-- The `cs`th element of the CKM matrix. -/
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scoped[CKMMatrix] notation (name := cs_element) "[" V "]cs" => V.1 1 1
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/-- The `cb`th element of the CKM matrix. -/
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scoped[CKMMatrix] notation (name := cb_element) "[" V "]cb" => V.1 1 2
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/-- The `td`th element of the CKM matrix. -/
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scoped[CKMMatrix] notation (name := td_element) "[" V "]td" => V.1 2 0
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/-- The `ts`th element of the CKM matrix. -/
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scoped[CKMMatrix] notation (name := ts_element) "[" V "]ts" => V.1 2 1
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/-- The `tb`th element of the CKM matrix. -/
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scoped[CKMMatrix] notation (name := tb_element) "[" V "]tb" => V.1 2 2
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/-- The setoid of CKM matrices defined by phase shifts of fermions. -/
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instance CKMMatrixSetoid : Setoid CKMMatrix := ⟨PhaseShiftRelation, phaseShiftRelation_equiv⟩
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/-- The matrix obtained from `V` by shifting the phases of the fermions. -/
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@[simps!]
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def phaseShiftApply (V : CKMMatrix) (a b c d e f : ℝ) : CKMMatrix :=
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phaseShift a b c * ↑V * phaseShift d e f
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namespace phaseShiftApply
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/-- A CKM matrix is equivalent to a phase-shift of itself. -/
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lemma equiv (V : CKMMatrix) (a b c d e f : ℝ) :
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V ≈ phaseShiftApply V a b c d e f := by
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symm
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use a, b, c, d, e, f
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rfl
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/-- The `ud` component of the CKM matrix obtained after applying a phase shift. -/
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lemma ud (V : CKMMatrix) (a b c d e f : ℝ) :
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(phaseShiftApply V a b c d e f).1 0 0 = cexp (a * I + d * I) * V.1 0 0 := by
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simp only [Fin.isValue, phaseShiftApply_coe]
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rw [mul_apply, Fin.sum_univ_three]
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rw [mul_apply, mul_apply, mul_apply, Fin.sum_univ_three, Fin.sum_univ_three, Fin.sum_univ_three]
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simp only [Fin.isValue, cons_val', cons_val_zero, empty_val', cons_val_fin_one, vecCons_const,
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cons_val_one, head_cons, zero_mul, add_zero, cons_val_two, tail_cons, head_fin_const, mul_zero,
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tail_val', head_val']
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change _ + _ * _ * 0 = _
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rw [exp_add]
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ring_nf
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/-- The `us` component of the CKM matrix obtained after applying a phase shift. -/
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lemma us (V : CKMMatrix) (a b c d e f : ℝ) :
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(phaseShiftApply V a b c d e f).1 0 1 = cexp (a * I + e * I) * V.1 0 1 := by
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simp only [Fin.isValue, phaseShiftApply_coe]
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rw [mul_apply, Fin.sum_univ_three]
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rw [mul_apply, mul_apply, mul_apply, Fin.sum_univ_three, Fin.sum_univ_three, Fin.sum_univ_three]
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simp only [Fin.isValue, cons_val', cons_val_zero, empty_val', cons_val_fin_one, vecCons_const,
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cons_val_one, head_cons, zero_mul, add_zero, cons_val_two, tail_cons, mul_zero, zero_add,
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head_fin_const]
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rw [exp_add]
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ring_nf
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/-- The `ub` component of the CKM matrix obtained after applying a phase shift. -/
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lemma ub (V : CKMMatrix) (a b c d e f : ℝ) :
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(phaseShiftApply V a b c d e f).1 0 2 = cexp (a * I + f * I) * V.1 0 2 := by
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simp only [Fin.isValue, phaseShiftApply_coe]
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rw [mul_apply, Fin.sum_univ_three]
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rw [mul_apply, mul_apply, mul_apply, Fin.sum_univ_three, Fin.sum_univ_three, Fin.sum_univ_three]
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simp only [Fin.isValue, cons_val', cons_val_zero, empty_val', cons_val_fin_one, vecCons_const,
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cons_val_one, head_cons, zero_mul, add_zero, cons_val_two, tail_cons, mul_zero, head_fin_const,
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zero_add]
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rw [exp_add]
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ring_nf
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/-- The `cd` component of the CKM matrix obtained after applying a phase shift. -/
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lemma cd (V : CKMMatrix) (a b c d e f : ℝ) :
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(phaseShiftApply V a b c d e f).1 1 0= cexp (b * I + d * I) * V.1 1 0 := by
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simp only [Fin.isValue, phaseShiftApply_coe]
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rw [mul_apply, Fin.sum_univ_three]
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rw [mul_apply, mul_apply, mul_apply, Fin.sum_univ_three, Fin.sum_univ_three, Fin.sum_univ_three]
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simp only [Fin.isValue, cons_val', cons_val_zero, empty_val', cons_val_fin_one, vecCons_const,
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cons_val_one, head_fin_const, zero_mul, head_cons, zero_add, cons_val_two, tail_cons, add_zero,
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mul_zero, tail_val', head_val']
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change _ + _ * _ * 0 = _
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rw [exp_add]
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ring_nf
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/-- The `cs` component of the CKM matrix obtained after applying a phase shift. -/
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lemma cs (V : CKMMatrix) (a b c d e f : ℝ) :
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(phaseShiftApply V a b c d e f).1 1 1 = cexp (b * I + e * I) * V.1 1 1 := by
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simp only [Fin.isValue, phaseShiftApply_coe]
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rw [mul_apply, Fin.sum_univ_three]
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rw [mul_apply, mul_apply, mul_apply, Fin.sum_univ_three, Fin.sum_univ_three, Fin.sum_univ_three]
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simp only [Fin.isValue, cons_val', cons_val_zero, empty_val', cons_val_fin_one, vecCons_const,
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cons_val_one, head_fin_const, zero_mul, head_cons, zero_add, cons_val_two, tail_cons, add_zero,
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mul_zero]
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rw [exp_add]
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ring_nf
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/-- The `cb` component of the CKM matrix obtained after applying a phase shift. -/
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lemma cb (V : CKMMatrix) (a b c d e f : ℝ) :
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(phaseShiftApply V a b c d e f).1 1 2 = cexp (b * I + f * I) * V.1 1 2 := by
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simp only [Fin.isValue, phaseShiftApply_coe]
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rw [mul_apply, Fin.sum_univ_three]
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rw [mul_apply, mul_apply, mul_apply, Fin.sum_univ_three, Fin.sum_univ_three, Fin.sum_univ_three]
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simp only [Fin.isValue, cons_val', cons_val_zero, empty_val', cons_val_fin_one, vecCons_const,
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cons_val_one, head_fin_const, zero_mul, head_cons, zero_add, cons_val_two, tail_cons, add_zero,
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mul_zero]
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rw [exp_add]
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ring_nf
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/-- The `td` component of the CKM matrix obtained after applying a phase shift. -/
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lemma td (V : CKMMatrix) (a b c d e f : ℝ) :
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(phaseShiftApply V a b c d e f).1 2 0= cexp (c * I + d * I) * V.1 2 0 := by
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simp only [Fin.isValue, phaseShiftApply_coe]
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rw [mul_apply, Fin.sum_univ_three]
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rw [mul_apply, mul_apply, mul_apply, Fin.sum_univ_three, Fin.sum_univ_three, Fin.sum_univ_three]
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simp only [Fin.isValue, cons_val', cons_val_zero, empty_val', cons_val_fin_one, vecCons_const,
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cons_val_two, tail_val', head_val', cons_val_one, head_cons, tail_cons, head_fin_const,
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zero_mul, add_zero, mul_zero]
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change (0 * _ + _) * _ + (0 * _ + _) * 0 = _
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simp only [Fin.isValue, zero_mul, zero_add, mul_zero, add_zero]
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rw [exp_add]
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ring_nf
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/-- The `ts` component of the CKM matrix obtained after applying a phase shift. -/
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lemma ts (V : CKMMatrix) (a b c d e f : ℝ) :
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(phaseShiftApply V a b c d e f).1 2 1 = cexp (c * I + e * I) * V.1 2 1 := by
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simp only [Fin.isValue, phaseShiftApply_coe]
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rw [mul_apply, Fin.sum_univ_three]
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rw [mul_apply, mul_apply, mul_apply, Fin.sum_univ_three, Fin.sum_univ_three, Fin.sum_univ_three]
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simp only [Fin.isValue, cons_val', cons_val_zero, empty_val', cons_val_fin_one, vecCons_const,
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cons_val_two, tail_val', head_val', cons_val_one, head_cons, tail_cons, head_fin_const,
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zero_mul, add_zero, mul_zero, zero_add]
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change (0 * _ + _) * _ = _
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rw [exp_add]
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ring_nf
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/-- The `tb` component of the CKM matrix obtained after applying a phase shift. -/
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lemma tb (V : CKMMatrix) (a b c d e f : ℝ) :
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(phaseShiftApply V a b c d e f).1 2 2 = cexp (c * I + f * I) * V.1 2 2 := by
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simp only [Fin.isValue, phaseShiftApply_coe]
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rw [mul_apply, Fin.sum_univ_three]
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rw [mul_apply, mul_apply, mul_apply, Fin.sum_univ_three, Fin.sum_univ_three, Fin.sum_univ_three]
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simp only [Fin.isValue, cons_val', cons_val_zero, empty_val', cons_val_fin_one, vecCons_const,
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cons_val_two, tail_val', head_val', cons_val_one, head_cons, tail_cons, head_fin_const,
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zero_mul, add_zero, mul_zero, zero_add]
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change (0 * _ + _) * _ = _
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rw [exp_add]
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ring_nf
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end phaseShiftApply
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/-- The absolute value of the `(i,j)`th element of `V`. -/
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@[simp]
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def VAbs' (V : unitaryGroup (Fin 3) ℂ) (i j : Fin 3) : ℝ := Complex.abs (V i j)
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/-- If two CKM matrices are equivalent (under phase shifts), then their absolute values
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are the same. -/
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lemma VAbs'_equiv (i j : Fin 3) (V U : CKMMatrix) (h : V ≈ U) :
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VAbs' V i j = VAbs' U i j := by
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simp only [VAbs']
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obtain ⟨a, b, c, e, f, g, h⟩ := h
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rw [h]
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simp only [Submonoid.coe_mul, phaseShift_coe_matrix]
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rw [mul_apply, Fin.sum_univ_three]
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rw [mul_apply, Fin.sum_univ_three]
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rw [mul_apply, mul_apply, Fin.sum_univ_three, Fin.sum_univ_three]
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simp only [phaseShiftMatrix, Fin.isValue, cons_val', cons_val_zero, empty_val', cons_val_fin_one,
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vecCons_const, cons_val_one, head_cons, zero_mul, add_zero, cons_val_two, tail_cons,
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head_fin_const, mul_zero]
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fin_cases i <;> fin_cases j <;>
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simp only [Fin.zero_eta, Fin.isValue, cons_val_zero, zero_mul, add_zero, mul_zero,
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_root_.map_mul, abs_exp, mul_re, I_re, ofReal_re, I_im, ofReal_im, sub_self, Real.exp_zero,
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one_mul, mul_one, Fin.mk_one, cons_val_one, head_cons, zero_add, head_fin_const,
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Fin.reduceFinMk, cons_val_two, Nat.succ_eq_add_one, Nat.reduceAdd, tail_cons,
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tail_val', head_val']
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all_goals change Complex.abs (0 * _ + _) = _
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all_goals simp [Complex.abs_exp]
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/-- The absolute value of the `(i,j)`th any representative of `⟦V⟧`. -/
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def VAbs (i j : Fin 3) : Quotient CKMMatrixSetoid → ℝ :=
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Quotient.lift (fun V => VAbs' V i j) (VAbs'_equiv i j)
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/-- The absolute value of the `ud`th element of a representative of an equivalence class of
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CKM matrices. -/
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@[simp]
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abbrev VudAbs := VAbs 0 0
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/-- The absolute value of the `us`th element of a representative of an equivalence class of
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CKM matrices. -/
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@[simp]
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abbrev VusAbs := VAbs 0 1
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/-- The absolute value of the `ub`th element of a representative of an equivalence class of
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||
CKM matrices. -/
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@[simp]
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abbrev VubAbs := VAbs 0 2
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||
|
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/-- The absolute value of the `cd`th element of a representative of an equivalence class of
|
||
CKM matrices. -/
|
||
@[simp]
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||
abbrev VcdAbs := VAbs 1 0
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||
|
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/-- The absolute value of the `cs`th element of a representative of an equivalence class of
|
||
CKM matrices. -/
|
||
@[simp]
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||
abbrev VcsAbs := VAbs 1 1
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||
|
||
/-- The absolute value of the `cb`th element of a representative of an equivalence class of
|
||
CKM matrices. -/
|
||
@[simp]
|
||
abbrev VcbAbs := VAbs 1 2
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||
|
||
/-- The absolute value of the `td`th element of a representative of an equivalence class of
|
||
CKM matrices. -/
|
||
@[simp]
|
||
abbrev VtdAbs := VAbs 2 0
|
||
|
||
/-- The absolute value of the `ts`th element of a representative of an equivalence class of
|
||
CKM matrices. -/
|
||
@[simp]
|
||
abbrev VtsAbs := VAbs 2 1
|
||
|
||
/-- The absolute value of the `tb`th element of a representative of an equivalence class of
|
||
CKM matrices. -/
|
||
@[simp]
|
||
abbrev VtbAbs := VAbs 2 2
|
||
|
||
namespace CKMMatrix
|
||
open ComplexConjugate
|
||
|
||
section ratios
|
||
|
||
/-- The ratio of the `ub` and `ud` elements of a CKM matrix. -/
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||
def Rubud (V : CKMMatrix) : ℂ := [V]ub / [V]ud
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||
|
||
/-- The ratio of the `ub` and `ud` elements of a CKM matrix. -/
|
||
scoped[CKMMatrix] notation (name := ub_ud_ratio) "[" V "]ub|ud" => Rubud V
|
||
|
||
/-- The ratio of the `us` and `ud` elements of a CKM matrix. -/
|
||
def Rusud (V : CKMMatrix) : ℂ := [V]us / [V]ud
|
||
|
||
/-- The ratio of the `us` and `ud` elements of a CKM matrix. -/
|
||
scoped[CKMMatrix] notation (name := us_ud_ratio) "[" V "]us|ud" => Rusud V
|
||
|
||
/-- The ratio of the `ud` and `us` elements of a CKM matrix. -/
|
||
def Rudus (V : CKMMatrix) : ℂ := [V]ud / [V]us
|
||
|
||
/-- The ratio of the `ud` and `us` elements of a CKM matrix. -/
|
||
scoped[CKMMatrix] notation (name := ud_us_ratio) "[" V "]ud|us" => Rudus V
|
||
|
||
/-- The ratio of the `ub` and `us` elements of a CKM matrix. -/
|
||
def Rubus (V : CKMMatrix) : ℂ := [V]ub / [V]us
|
||
|
||
/-- The ratio of the `ub` and `us` elements of a CKM matrix. -/
|
||
scoped[CKMMatrix] notation (name := ub_us_ratio) "[" V "]ub|us" => Rubus V
|
||
|
||
/-- The ratio of the `cd` and `cb` elements of a CKM matrix. -/
|
||
def Rcdcb (V : CKMMatrix) : ℂ := [V]cd / [V]cb
|
||
|
||
/-- The ratio of the `cd` and `cb` elements of a CKM matrix. -/
|
||
scoped[CKMMatrix] notation (name := cd_cb_ratio) "[" V "]cd|cb" => Rcdcb V
|
||
|
||
lemma Rcdcb_mul_cb {V : CKMMatrix} (h : [V]cb ≠ 0) : [V]cd = Rcdcb V * [V]cb := by
|
||
rw [Rcdcb]
|
||
exact (div_mul_cancel₀ (V.1 1 0) h).symm
|
||
|
||
/-- The ratio of the `cs` and `cb` elements of a CKM matrix. -/
|
||
def Rcscb (V : CKMMatrix) : ℂ := [V]cs / [V]cb
|
||
|
||
/-- The ratio of the `cs` and `cb` elements of a CKM matrix. -/
|
||
scoped[CKMMatrix] notation (name := cs_cb_ratio) "[" V "]cs|cb" => Rcscb V
|
||
|
||
/-- Multiplying the ratio of the `cs` by `cb` element of a CKM matrix by the `cb` element
|
||
returns the `cs` element, as long as the `cb` element is non-zero. -/
|
||
lemma Rcscb_mul_cb {V : CKMMatrix} (h : [V]cb ≠ 0) : [V]cs = Rcscb V * [V]cb := by
|
||
rw [Rcscb]
|
||
exact (div_mul_cancel₀ [V]cs h).symm
|
||
|
||
end ratios
|
||
|
||
end CKMMatrix
|
||
|
||
end
|